Medical diagnostic tests (or markers) are used to indicate whether a patient has a certain disease, but they typically can yield false positive or false negative results. Choosing a good diagnostic test is very important in medical practice; a typical and important example is the problem of choosing between two serum antibodies as markers of a certain type of cancer. This project will investigate statistical methods of comparison of two diagnostic markers under the assumption that the data, which consist of observations of two markers on each patient come from two bivariate normal distributions, one distribution for the cases and one for the controls. Efficient statistical tests for differences between the markers will be sought under various hypotheses on the parameters. These hypotheses will stipulate equality, for the two markers, of weighted averages of Receiver Operating Characteristic (ROC) curve values. Likelihood ratio tests will be investigated; if these prove intractable for some hypotheses, optimal tests of type C(a) will be considered. Once efficient statistical tests are found, their properties will be investigated. The distribution of the test statistics under the null hypotheses will be determined, at least approximately, where feasible. If this is not feasible, the distribution will be simulated using Monte Carlo methods, which will also be used to study the power functions of the test. The tests found will then be compared with others from the literature. A topic for future research, and which will be initiated in this project, is a sequential comparison of diagnostic markers based on the fixed-sample tests obtained in the above study.